Tripartite optomechanical entanglement via optical-dark-mode control

We propose how to generate a tripartite light-vibration entanglement by controlling an optical dark mode (ODM), which is induced by the coupling of two optical modes to a common vibrational mode. This ODM is decoupled from the vibration, and it can be controlled on demand by employing a synthetic gauge ﬁeld, which can enable efﬁcient switching between the ODM-unbreaking and ODM-breaking regimes. We ﬁnd that the tripartite optomechanical entanglement is largely suppressed in the ODM-unbreaking regime, but it is signiﬁcantly enhanced in the ODM-breaking regime. In particular, the noise robustness of quantum entanglement in the ODM-breaking regime can be more than twice than that in the ODM-unbreaking regime. This study offers a method for protecting and enhancing fragile quantum resources and for constructing noise-tolerant and dark-mode-immune quantum processors and entangled networks.

Beyond bipartite quantum entanglement, multipartite entangled states can offer a more fundamental resource for a wide range of quantum information processing tasks [43][44][45], and thus, they can be used for future quantum technologies, such as quantum internet and programmable quantum networks [43][44][45].However, based on cavity optomechanics, the generation of multipartite optomechanical entanglement (e.g., quantum entanglement of multiple cavity-field modes and a common vibrational mode), and, in particular, its protection in practical devices have not yet been fully revealed.
In recent years, much attention has been drawn to optical dark modes (ODMs) [46][47][48][49][50], which are special coherent superpositions of, e.g., two optical modes coupled to a common vibration in optomechanical systems.Owing to their unique characteristics in decoupling from the mechanical vibration, the ODM can be employed for protecting systems from mechanical dissipation [46], enable high fidelity conversion of optical fields or quantum states [47][48][49], and realize efficient routing and switching of photons of different wavelengths [50].In light of the above potential applications and unique characteristics, it is natural to study the influence of the ODM on light-vibration quantum effects, e.g., optomechanical entanglement.
In this paper, we propose how to generate the tripartite light-vibration entanglement by controlling the ODM in a three-mode closed-loop optomechanical system, and reveal its strong robustness against thermal noise.This is realized by utilizing a synthetic gauge field, which provides a possibility of fully controllable switching between the optical-darkmode-unbreaking (ODMU) and optical-dark-mode-breaking (ODMB) regimes.Specifically, driving the two cavities with phase-correlated lasers results in a synthetic magnetic flux, which induces a synthetic gauge field.This synthetic gauge field with a topological phase has, respectively, been theoretically proposed [51][52][53][54] and experimentally demonstrated [55][56][57][58][59][60][61] in closed-loop optomechanical systems.We find that in the ODMU regime, tripartite light-vibration entanglement is strongly suppressed by the ODM; while in the ODMB regime, it is significantly enhanced.The physical origin behind these phenomena is that the ODM is decoupled from the vibrational mode, which leads to the suppression of tripartite optomechanical entanglement.However, the breaking of the ODM enhances tripartite entanglement.These results show a clear inspiration for enhancing tripartite optomechanical entanglement by controlling the ODM.In particular, the resulting ODMB entanglement is noisetolerant, which is up to twice as strong as that in the ODMU regime.Our findings pave a way towards engineering and protecting fragile quantum correlations from the ODM and thermal noise, and offer a possibility of integrating quantum processors.
The rest of the paper is organized as follows.In Sec.II, we present the Hamiltonian of a three-mode closed-loop optomechanical system, derive the corresponding Langevin equations, and obtain their solutions.In Sec.III, we derive the minimum residual contangle, which can be used to quantify the tripartite quantum entanglement, and analyze how to control an ODM.In Sec.IV, we study the tripartite lightmotion entanglement by controlling the ODM.In Sec.V, we present some discussions.Finally, we give a brief conclusion in Sec.VI.

II. HAMILTONIAN, LANGEVIN EQUATIONS, AND THEIR SOLUTIONS
We consider a three-mode closed-loop optomechanical system, where two optical modes are optomechanically coupled to a common mechanical resonator, as shown in Fig. 1(a).The two cavity-field modes are coupled to each other via a photon-tunneling interaction with strength J.To manipulate the optical and mechanical degrees of freedom in the system, we assume that the jth cavity is subjected to a monochromatic driving field, with frequency ω L, j and amplitude j = 2κ j P L, j / hω L, j where κ j is the cavity-field decay rate and P L, j is the laser power.In the interaction picture, shutting all the time dependence due to H 0 = 2 j=1 ω L, j a † j a j , the Hamiltonian of the system is (with h = 1) where c, j = ω c, j − ω L, j is the driving detuning of the jth cavity.The operators a j (a † j ) and b (b † ) are the annihilation (creation) operators of the jth optical mode and the vibrational mode, respectively.The g j term describes the light-vibration interactions between the jth optical mode and the vibrational mode.The cavity-field driving, which are applied to the jth cavity, is denoted by the j terms.
By considering damping and noise effects in the system, the evolution of this system can be described by the quantum Langevin equations where a j,in and b in are the noise operators of the jth cavity mode and the vibrational mode, respectively.These noise operators have zero mean values and satisfy the standard correlation functions: where n = [exp( hω m /k B T ) − 1] −1 is the thermal phonon number of the vibration, T denotes the temperature of the thermal reservoir associated with the vibration, and k B is the Boltzmann constant.We consider the strong-driving regime, so that the average photon numbers in the two cavities are sufficiently large.Then, a linearization procedure can be used to simplify the physical model.To this end, we express the operators in Eq. ( 2) as a sum of their steady-state mean values and quantum fluctuations, namely: o = o ss + δo, for operators o = a j , a † j , b, and b † .By separating the classical motion from quantum fluctuations, the linearized equations of motion for quantum fluctuations can be written as where j = c, j + g j (β + β * ) is the normalized driving detuning of the jth cavity field, and the linearized optomechanical coupling strength G j , with phase θ j , between the jth cavity-field mode and the vibrational mode is given by where the steady-state average values of the dynamical variables are ).Note that a synthetic gauge field with a modulation phase has recently been realized using three-mode closed-loop optomechanical systems [61], in which the appropriate spatially dependent hopping phases are realized through external drives [55][56][57][58][59][60][61].
By defining the mechanical and optical quadratures and the corresponding Hermitian input-noise operators we obtain a compact form of the linearized equations of quantum fluctuations where T is the noise operator vector, and is the coefficient matrix, where G j± = G * j ± G j .The formal solution of the linearized Langevin equations in ( 6) is given by where M(t ) = exp(At ).

A. Minimum residual contangle and entanglement monogamy
For studying the tripartite quantum entanglement, we focus on calculating the steady-state value of the covariance matrix V, which is defined by the matrix elements for k, l = 1, . . ., 6.Under the stability condition, the covariance matrix V fulfills the Lyapunov equation where [62,63] can be used to quantify a tripartite quantum entanglement, defined as and is referred here as a tripartite entanglement measure, where E u|v τ is the contangle of subsystems of u and v (v contains one or two modes), and r, s, t ≡ (d 1 , d 2 , c) denotes all the permutations of the three mode indexes [62].Note that E u|v τ is defined as the squared logarithmic negativity [62][63][64][65], and it is a proper entanglement monotone.
The residual contangle satisfies the monogamy of quantum entanglement which is based on the Coffman-Kundu-Wootters monogamy inequality [64].Entanglement monogamy is the fundamental principle that quantum entanglement cannot be freely shared between three subsystems (in general, among arbitrarily many systems).
Here, E r|s|t τ > 0 indicates that a tripartite quantum entanglement is generated.Note that all the parameters in the following calculations satisfy the stability condition, which is derived using the Routh-Hurwitz criterion [66].In addition, we have confirmed that only a single stable solution exists and our system has no bistability in all our simulations.
In light of the above achievements in realizing the synthetic gauge field using the three-mode closed-loop optomechanical platforms [61], we below employ this synthetic gauge field to control the ODM in these systems, so that the tripartite light-vibration entanglement can be flexibly controlled.Specifically, we here consider the following two cases: (i) In the absence of the synthetic gauge field, i.e., J = 0, the two optical modes can form an optical bright mode A + and an optical dark mode A − , which are given by where Then, the Hamiltonian in Eq. ( 12) can be rewritten as where ]/G 2 0 and the coupling strengths are We can see from Eqs. ( 14) and ( 15) that when 1 = 2 , the mode A − is decoupled from the system due to G − = 0, and it becomes an ODM.Owing to the decoupling of the ODM from the vibrational mode, the generation of the tripartite light-vibration entanglement is largely suppressed.
(ii) Since the ODM leads to the suppression of the tripartite entanglement generation, it is natural to ask the question whether one can break this ODM to further generate purer tripartite optomechanical entanglement.To this end, a synthetic magnetism is introduced in the system for controlling the ODM, i.e., J = 0. Specifically, by introducing two superposition-optical modes associated with the synthetic magnetism Ã+ and Ã− , defined by the Hamiltonian in Eq (12) becomes where , and the redefined coupling strengths are given by In Fig. 1(b), the redefined coupling strengths G± are plotted as a function of the modulation phase θ .It shows that only at θ = nπ (i.e., in the ODMU regime), Ã+ [for an odd n, G+ = 0 (see blue symbols)] or Ã− [for an even n, G− = 0 (see red symbols)] becomes the ODM.A counterintuitive coupling of the ODM to the vibrational mode can be achieved by tuning θ = nπ (i.e., in the ODMB regime), which means the breaking of the ODM [see Fig. 1(b)].The underlining physical mechanism is that the constructive or destructive interference between the two coupling paths, by modulating the phase θ , enables a flexible switch between the ODMU and ODMB regimes.
To provide a clearer interpretation of our results, we here derive various approximate analytical expressions for the redefined coupling strengths in the symmetric-detuning and symmetric-coupling regimes, as well as in the resolvedsideband regime.In the symmetric-detuning ( j=1,2 = ) and symmetric-coupling (G 0 j=1,2 = G) cases, the redefinedcoupling strengths G± in Eq. ( 18) become G± ≈ G(1 ± e ∓iθ )/ √ 2. One can clearly see that the redefined-coupling strengths G± strongly depend on the modulation phase θ .Furthermore, by considering the case of j=1,2 = , κ j=1,2 = κ, g 01 = g 02 = g, and J ω m , , in the resolved-sideband regime (i.e., κ ω m ), we obtain the approximate analytical expressions of the redefined coupling strengths as: G± ≈ g (1 ± e ∓iθ )/ √ 2. It can be intuitively seen that in the resolved-sideband regime, (i) the DMB is governed by the modulation phase θ and (ii) the strength of the redefinedcoupling strengths can be enhanced by tuning the amplitude of the driving lasers applied to the optical cavities.These observations based on our analytical approximate results are well matched with the numerical simulations based on the exact results.
Here it is worth presenting some discussions on the reason for choosing the values of the system parameters.In realistic systems, the strength of the linearized effective optomechanical coupling should be much smaller than mechanical resonance frequency ω m .In addition, the photon-tunneling interaction between the two cavities can be realized by optical backscattering [34,[59][60][61].This backscattering of the photons is induced by the surface roughness and material defects in practical devices.Therefore the value of the photon-tunneling coupling strength used in our simulations should be of the same order of the decay rates of the cavity-field modes.These assumed values of parameters should be accessible under the near-future experimental conditions.In this work, to obtain a significative amount of quantum entanglement, we have made a careful analysis in a wide parameter range by combining theoretical [24] and experimental [38][39][40][41][42] works, and we have assumed experimentally feasible conditions in our manuscript.

IV. TRIPARTITE LIGHT-MOTION ENTANGLEMENT
In the above section, we have derived the minimum residual contangle, and have analyzed how to control the ODM in the three-mode optomechanical system.Now we study in detail the tripartite optomechanical entanglement by controlling the ODM.
In Fig. 2(a), we plot the tripartite light-vibration entanglement, quantified by the minimum residual contangle E r|s|t τ , as a function of the driving detunings j = of the two cavities, when the system operates in both ODMU (J = 0, see the blue curve) and ODMB (J/ω m = 0.2 and θ = π/2, see the red curve) regimes.We find that in the ODMB regime, the tripartite entanglement is nearly twice as large as that in the ODMU regime.The underlying physical mechanism is that the ODM is decoupled from the vibration, and this results in the suppression of the tripartite optomechanical entanglement.However, the breaking of the ODM leads to the enhancement in quantum entanglement.These results show a clear inspiration for enhancing tripartite optomechanical entanglement via the control of the ODM.
Moreover, we find from Fig. 2(b) that, when turning off the synthetic gauge field (i.e., J = 0, see the blue curve), the tripartite optomechanical entanglement is strongly suppressed by the ODM, corresponding to the emergence of the dip [see the blue curve in Fig. 2(b)].In particular, this ODM effect can work in a wider driving detuning range, i.e., 2 / 1 > 0.5.In contrast to this, once turning on the synthetic gauge field (i.e., J = 0 and θ = nπ ), the dip becomes a peak [see the red curve in Fig. 2(b)], and this indicates that the tripartite light-vibration entanglement is significantly enhanced due to breaking the ODM.In particular, the maximum amount of the resulting entanglement in the ODMB regime is larger than that in the ODMU case.
To determine how significant enhancement in the tripartite optomechanical entanglement can be reached, we here introduce an entanglement-amplification factor , which is defined as Based on Eq. ( 19), we show the effect of the parameters J and θ of the synthetic magnetism on the entanglementamplification factor in Fig. 3.We find that the tripartite optomechanical entanglement can be significantly amplified by tuning the synthetic magnetism parameters.For example, when the synthetic magnetism is off (i.e., when J = 0), no amplification for the tripartite entanglement can be observed (i.e., = 1, see the white horizontal dashed line).However, when the synthetic magnetism is on, the amplification of the tripartite entanglement emerges, and even the entanglement amplification factor can increase up to = 2.24.When J/ω m → 0.4 and θ → nπ for an integer n, we obtain 0 < < 1.This is because the optical backscattering losses in practical devices [34,61] strongly suppress the entanglement generation.Physically, the backscattering of photons can be induced by various imperfections of devices, such as surface roughness and material defects, as described by, e.g., the photon-tunneling coupling J [34,61].
In the above section, we have showed that a significant enhancement in the tripartite optomechanical entanglement can be achieved via the ODM control.It is natural to ask the question: can we further improve the tripartite light-vibration entanglement by tuning the other parameters of the system?
For further elucidating this problem, we plot the tripartite light-vibration entanglement in Fig. 4(a), quantified by the minimum residual contangle E r|s|t τ , as functions of the optomechanical coupling strengths G 01 and G 02 in the ODMB regime.We see that the tripartite optomechanical entanglement can be significantly enhanced by increasing G 0 j , and that the maximal amount of the tripartite entanglement can be observed for 0.2 G 0 j /ω m 0.3.Moreover, in Fig. 4  plot the tripartite optomechanical entanglement measure E r|s|t τ as a function of the light-motion coupling strengths G 0 j in both ODMU (blue solid curve) and ODMB (red dashed curve) regimes.We find that in both ODMU and ODMB regimes, the tripartite optomechanical entanglement could be further enhanced by increasing the optomechanical coupling strengths G 0 j .In particular, we show that the tripartite optomechanical entanglement in the ODMB regime is larger than that in the ODMU regime (i.e., E r|s|t τ,ODMB > E r|s|t τ,ODMU ).In Fig. 4(c), the tripartite optomechanical entanglement measure E r|s|t τ is plotted as functions of the decay rates κ 1 and κ 2 of the two cavities, when the system works in the ODMB regime.For clearly investigating the influence of the sideband-resolution condition on the tripartite optomechanical entanglement, the mechanical frequency ω m is chosen as a frequency scale.This demonstrates that the phonon sideband can be well resolved from the cavity-emission spectrum when κ j /ω m 1, corresponding to the resolved-sideband limit [10][11][12][13].In particular, in this resolved-sideband regime, i.e., κ j /ω m 1, We find that the tripartite optomechanical entanglement becomes much larger for a smaller cavity-field decay rate κ j .
To further elucidate this observation, we plot the tripartite optomechanical entanglement measure E r|s|t τ as a function of the cavity-field decay rate κ j , in both ODMU (blue solid curve) and ODMB (red dashed curve) regimes, as shown in Fig. 4(d).We can see that in these two regimes, the phonon sidebands can be well resolved (i.e., κ j /ω m 1, see the blueshaded area), and that the tripartite quantum entanglement can be significantly improved with decreasing κ j .However, in the unresolved-sideband regime κ j /ω m > 1, the tripartite entanglement becomes much worse for a larger cavity-field decay rate κ j (see the yellow-shaded area).Despite this, when κ j /ω m 1, the amount of the tripartite light-vibration entanglement in the ODMB regime is much larger than that in the ODMU regime (i.e., E r|s|t τ,ODMB > E r|s|t τ,ODMU ).These findings offer a different method to enhance fragile quantum resources by appropriately designing the cavities.
Our ODMB mechanism paves a feasible way to enhance fragile quantum resources via the control of an optical dark mode, and it can be used for constructing the noise-tolerant quantum processors and quantum networks.To study the noise robustness of the resulting tripartite optomechanical entanglement, we plot the minimum residual contangle E r|s|t τ as a function of the thermal phonon number n of the vibration, when the system operates in both ODMB and ODMU regimes, as shown in Fig. 5.We find that for the ODMU regime, the tripartite optomechanical entanglement emerges only for the thermal phonon number n 1500 (see the blue symbols).However, for the ODMB regime, it can persist for the thermal phonon number near n = 3000 (see the red symbols), which is twice that of the ODMU case.In particular, we find that with the increase of the thermal phonon number n, the tripartite light-vibration entanglement in the ODMB regime is always much lager than that in the ODMU regime.Our findings provide a way for enhancing and protecting fragile quantum resources from thermal noise and ODM, and pave a way towards noise-tolerant large-scale entanglement networks.

V. DISCUSSIONS
In this section, we show some discussions on possible experimental realizations of our model, and on the tripartite entanglement measures and the importance of our measure of entanglement.

A. Discussions on possible experimental realizations
In this section, some discussions are presented for showing the experimental realization of the photon-tunneling coupling for our model and related ones.Currently, the photon-hopping interaction between the two cavity fields has been experimentally implemented in a microsphere optomechanical cavity system [59][60][61]67], where a microresonator supports a pair of degenerate counterclockwise and clockwise traveling-wave whispering-galley modes.The resonant frequencies of these modes are modulated by a radial breathing (vibrational) mode, which changes the circumference of a microsphere.Then, the optomechanical coupling is induced between the radial breathing mode and the counterclockwise (clockwise) mode.Note that the photon-tunneling interaction between the counterclockwise and clockwise optical modes is realized due to optical backscattering, and, in experiments, this backscattering of photons can be induced by the surface roughness and material defects in practical devices [59][60][61].By utilizing counterclockwise and clockwise optical modes and a vibrational mode, we can induce a precisely controllable synthetic gauge field by just tuning the phases of the external driving lasers [61].
In addition, based on membrane-in-the-middle configuration optomechanical-cavity systems [68], we can realize the photon-tunneling coupling between the two cavity fields and, then, implement the loop-coupled optomechanical system in experiments in an optomechanical-cavity system.In the membrane-in-the-middle configuration, there exist two cavity-field modes (the left and right subcavity modes), which are coupled to each other via a photon-tunneling interaction.The vibration of the mechanical membrane modulates the resonant frequencies of the two subcavity modes and, as a result, the radiation-pressure coupling is induced between the left (right) subcavity mode and this vibrational mode.
Actually, this nontrivial phase can also be realized by using multiple tones based on cavity optomechanical systems [58,69,70].For example, in a four-mode loop-coupled optomechanical configurations [58,69], two cavity modes are coupled indirectly via two nondegenerate vibrational modes, and the system is driven by four tones in the well-resolved sidebands.In this case, the nontrivial phase of the system can be induced by tuning the phases of the driving tones.In addition, using the standard optomechanical interactions can lead to a nontrivial phase in a three-mode optomechanical system [70], where two nondegenerate resonators are coupled to a common cavity.In that system, the cavity is driven by four tones, and this nontrivial phase can be tuned in situ simply via the phases of the driving tones applied to the cavity.

B. Discussions on tripartite entanglement measures and the importance of our measure of entanglement
Here we present some discussions on tripartite entanglement measures and on the importance of the measure of entanglement applied by us.TABLE I. Five disjoint classes of three-mode (tripartite) states concerning their bipartite and tripartite entanglement based on the classification of Ref. [71].Here "+" indicates that a given is nonzero.Examples of states ρ ≡ ρ ABC of each class can be generated by our optomechanical system for the parameters, which are specified in the last column.Here A (i.e., an example for class 1) includes the GHZ-like states of Ref. [72]; B (i.e., an example for class 2) is the two-mode squeezed vacuum in the first two modes and the vacuum in the third mode; C (i.e., an example for class 3) includes separable states with respect to two of the three bipartitions but inseparable with respect to the third bipartition; D (i.e., an example for class 4) includes separable states with respect to all three bipartitions but cannot be written as a mixture of tripartite product states; E (i.e., an example for class 5) includes the vacuum state in all three modes.Note that tripartite entanglement corresponds to three-mode biseparability (class 4).

Class
State ρ , and E C|(AB) ρ ) entanglement for a given three-mode state ρ ≡ ρ ABC such that they are zero if and only if ρ can be decomposed, respectively, as follows: These measures of entanglement can be chosen in various ways, and ours are specified below.Note that the above singlemode (ρ A i ,...) and two-mode (ρ AB i ,...) mixed states can be replaced by pure states (as in, e.g., Ref. [71]) without loosing any generality of these decompositions.
Following the classification introduced in Ref. [71] (and then applied to many specific systems, including optomechanical ones in, e.g., Ref. [73]), one can distinguish five different classes of entanglement, which are listed in Table I for a system composed of three modes (or parties, say A, B, and C).Note class 4 of three-mode biseparable states, which are separable with respect to all three bipartitions, but cannot be written as a mixture of tripartite product states, given in Eq. (23).These are usually referred to as tripartite entangled states, and play a central role in this paper.
In particular, various measures of entanglement can be used to properly classify bipartite and tripartite entanglement of classes of states listed in Table I.Concerning measures of bipartite entanglement, one can choose among dozens of measures, but for quantifying tripartite entanglement the choice of measures is much more limited [74].In addition to the minimum residual contangle introduced in Ref. [62], as studied by us, one could apply, in principle, a finite-dimensional multipartite concurrence introduced in Refs.[75,76] in the limit of large-dimensional modes.That measure has also an intuitive physical interpretation, but it is much more difficult to be calculated for systems of a large dimension (note ours is infinite-dimensional), because it requires finding numerically an infimum over all possible decompositions of a given mixed state.
We note that tripartite entanglement optomechanical systems was studied already two decades ago (see, e.g., Refs.[73,77,78]).In particular, Ref. [78] discusses how to generate bipartite and tripartite entanglement in a system composed of three subsystems (modes, denoted here as A, B, C): a micromechanical resonator and two output optical fields.Specifically, (i) purely optical bipartite entanglement between two output modes (of the two-mode reduced system) and (ii) fully tripartite optomechanical entanglement (corresponding to class 1 in Table I).Bipartite entanglement is quantified by the (logarithmic) negativity corresponding to the negative eigenvalue of the partially transposed (PT) two-mode-reduced matrices.Tripartite entanglement is revealed and quantified analogously by the minimum eigenvalues for the PT threemode matrices with respect to each of the three modes.This corresponds to applying bipartite PT-based entanglement measures for all the three different bipartitions of the ABC system, i.e., A|(BC), B|(CA), and C|(AB).It is found, for some ranges of the system parameters, that these eigenvalues can be all negative, showing that their system can be fully tripartite entangled [73].Clearly, tripartite entanglement (corresponding to Class 4) occurs if a tripartite state ρ exhibits both (i) E A|(BC) ρ = E B|(CA) ρ = E C|(AB) ρ = 0 and (ii) E A|B|C ρ > 0. By checking conditions (i) without testing conditions (ii) (i.e., without calculating E A|B|C ρ ), one cannot say whether a given state ρ belongs to Class 4 or Class 5 of completely separable states.However, by calculating E A|B|C ρ via the minimum residual contangle, we are able to distinguish these two cases for our model.
In fact, the contangle (and its absolute value) can be useful for various quantum information applications.Here some discussions are shown to clearly explain the importance of the measure of entanglement applied by us.In this work, the main reason for us of applying the minimum residual contangle, as a measure of a tripartite entanglement, is its relation to entanglement monogamy in addition to its simple computability.In fact, the measure was introduced in Ref. [62], as a continuous-variable analog of the famous Coffman-Kundu-Wootters monogamy inequality [64] designed for testing the monogamy of entanglement of discrete systems.Entanglement monogamy describes one of the most fundamental properties of quantum entanglement, which means that entanglement cannot be freely shared between arbitrarily many parties (in our case, three modes).The monogamy of entanglement has many fundamental applications (for a review see Ref. [79]).It is especially useful in proving the security of quantum cryptosystems (see, e.g., Refs.[80,81]).But the interest in multipartite entanglement and its monogamy is not limited to quantum cryptology, quantum communication, and quantum computing [82].For example, an apparent violation of entanglement monogamy, which is referred to as polyamory and studied in, e.g., black holes [83], would lead to major paradoxes of quantum mechanics and cosmology.

VI. CONCLUSIONS
We have showed how to realize a tripartite optomechanical entanglement via the control of the ODM, which is formed by two optical modes coupled to a common vibrational mode.The tripartite light-vibration entanglement can be significantly enhanced via the ODMB mechanism, without which it is strongly suppressed.In particular, the noise robustness of the tripartite quantum entanglement in the ODMB regime can be even twice as large as that in the ODMU regime.This study can enable constructing large-scale entanglement networks with the dark-mode immunity and the noise tolerance, and also open up a range of exciting opportunities for quantum information processing and quantum metrology protected against the ODM.

1 κFIG. 1 .
FIG.1.(a) Schematics of a three-mode closed-loop optomechanical system.Two cavities a j=1,2 (resonance frequencies ω c, j and decay rates κ j ) are coupled to a common vibration b (resonance frequency ω m and decay rate γ m ) via radiation-pressure couplings g j .The two cavities a 1 and a 2 are coupled to each other through a photon-tunneling interaction with strength J.A monochromatic driving field, with frequency ω L, j and field amplitude j , is introduced to drive the jth cavity.(b) In the polar coordinates, the redefined coupling strengths G± is shown versus the modulation phase θ.The ODM of the system emerges at θ = nπ (for an integer n), corresponding to the ODMU regime, and it can be broken ( G± = 0) by tuning θ = nπ , corresponding to the ODMB regime.The parameters used here are G 0 j /ω m = 0.1 and J/ω m = 0.1.

FIG. 3 .
FIG.3.Amplification factor of the tripartite optomechanical entanglement versus the photon-tunneling coupling J and the modulation phase θ under the optimal driving detunings j=1,2 /ω m = 0.45.Here the white dashed curve denotes the case of J = 0. Other parameters are the same as those in Fig.2.

FIG. 5 .
FIG. 5. Tripartite optomechanical entanglement measure E r|s|t τ as a function of the thermal phonon number n of the mechanical resonator in the ODMU (blue symbols) and ODMB (red symbols) regimes.Other parameters are the same as those in Fig. 2.